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Vector projection. The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as or a∥b.
Projection (linear algebra) Idempotent linear transformation from a vector space to itself. "Orthogonal projection" redirects here. For the technical drawing concept, see Orthographic projection. For a concrete discussion of orthogonal projections in finite-dimensional linear spaces, see Vector projection. The transformation P is the orthogonal ...
The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
Orthographic projection (also orthogonal projection and analemma) [a] is a means of representing three-dimensional objects in two dimensions.Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, [2] resulting in every plane of the scene appearing in affine transformation on the viewing surface.
The Gram–Schmidt process takes a finite, linearly independent set of vectors for k ≤ n and generates an orthogonal set that spans the same -dimensional subspace of as . The method is named after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. [1]
Distance from a point to a plane. In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane. It can be found starting with a change of variables that moves the origin to coincide with the given point then ...
This type of projection naturally generalizes to any number of dimensions n for the domain and k ≤ n for the codomain of the mapping. See Orthogonal projection, Projection (linear algebra). In the case of orthogonal projections, the space admits a decomposition as a product, and the projection operator is a projection in that sense as well.
Moore–Penrose inverse. In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. [1] It was independently described by E. H. Moore in 1920, [2] Arne Bjerhammar in 1951, [3] and Roger Penrose in 1955. [4]